Basic extreme value analysis

Classical approaches

1. Block maxima

2. Peaks over threshold

Block maxima

Largest in a block of observations

Peaks over threshold

Distribution over a threshold

Blocks of observations

\[\{X_1, X_2, ..., X_n \}\]

Block maxima

\[M_n = \text{max}\{X_1, X_2, ..., X_n \}\]

A little bit of theory

Extremal types theorem

If \(\text{Pr}(M_n \leq z)\) is well-behaved, it belongs to one of 3 families:

Generalized extreme value (GEV) distribution

\[G(z) = \text{exp}\Big\{- \Big[ 1 + \xi \Big( \dfrac{z - \mu}{\sigma} \Big) \Big]^{-1/\xi} \Big\}\]

defined over \(\{z: 1 + \xi (z - \mu)/\sigma \gt 0 \}\)

GEV parameters

\[G(z) = \text{exp}\Big\{- \Big[ 1 + \xi \Big( \dfrac{z - \mu}{\sigma} \Big) \Big]^{-1/\xi} \Big\}\]

• \(\mu:\) location, \(-\infty \lt \mu \lt \infty\)
• \(\sigma:\) scale, \(\sigma \gt 0\)
• \(\xi:\) shape, \(-\infty \lt \xi \lt \infty\)

Moving between families

The shape parameter determines the GEV family

• \(\xi = 0\): Gumbel
• \(\xi \gt 0\): Fréchet
• \(\xi \lt 0\): Weibull

Conventional applications

Observe maxima

Estimate GEV parameters

Infer probabilities, future extremes, etc.

Today’s activity

Observe Boulder Creek discharge data

Estimate GEV parameters

Infer return intervals for 2013 floods